Math 279, Semiclassical Analysis (Topics in PDE), Fall 2018

Semiclassical analysis is a version of microlocal analysis, which is the mathematical theory inspired by the classical/quantum and particle/wave correspondences in physics. It applies to a wide variety of partial differential equations. Some classical applications are to the wave, Schrödinger, and high-frequency Helmholtz equations. More recent applications include general relativity (e.g. the work of Hintz–Vasy on stability of Kerr–de Sitter black holes as solutions of Einstein equations) and dynamical zeta functions (a surprising recent development).

This course presents fundamental components of semiclassical analysis, including:

• Semiclassical quantization, which associates to a function a(x,ξ) of (x,ξ)∈R²ⁿ an operator Op_h(a) acting on functions on Rⁿ. In some sense this operator is obtained by replacing ξ by -ih∂_x, but this makes little sense for general functions a and the actual definition uses the Fourier transform instead. Operators of the form Op_h(a) are called semiclassical pseudodifferential operators. This class includes (semiclassical) differential operators but is much more versatile.
• Standard properties of semiclassical quantization: rules for computing products and adjoints of pseudodifferential operators, L² boundedness.
• Quantization on manifolds, where R²ⁿ is replaced by the cotangent bundle T^*M. The latter is naturally a symplectic manifold and semiclassical analysis makes natural use of some basic symplectic geometry such as Hamiltonian flows and the Darboux theorem.
• Microlocalization, namely using pseudodifferential operators to localize functions in both position and frequency and define their wavefront sets.
• Egorov's Theorem, which is a form of the classical/quantum correspondence principle. It implies that the wavefront set of the quantum evolution e^-itP/hu of a function u is obtained from the wavefront set of u as follows: WF_h(e^-itP/hu)=e^tH(WF_h(u)) where e^tH is the Hamiltonian flow of p and P=Op_h(p). This shows in particular the validity of the geometric optics approximation: at high frequencies, waves propagate along straight lines.

We will also study some applications of semiclassical analysis, such as:

• Weyl's Law for the asymptotic distribution of large eigenvalues of the Laplacian.
• Quantum ergodicity which shows equidistribution in mass of most eigenfunctions on manifolds with ergodic (chaotic) geodesic flows. This is a now-classical result in the field of quantum chaos which lies at the intersection of semiclassical analysis and dynamical systems.

Background for the course includes manifolds, distributions, and symplectic geometry. Deep knowledge of manifolds is not needed to take the course (many results are already interesting on Rⁿ) and distributions/symplectic geometry will be reviewed as needed during the course. As is common for a topics course, I will often defer to the book for technical details of some proofs.